Sampling Distributions

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Exercise: Use the same population of X described above and the samples of size 2 to create the
sampling distribution of the sample variance. This means you have to
(1) Construct all possible samples of size 2. (We have already done this.)
(2) Determine the probability of seeing each type of sample of size 2. (We have already done
this.)
(3) For each sample, compute the sample variance. For example, for the sample (1,4), we
compute
( xi  x ) 2 (1  2.5) 2  (4  2.5) 2

s 

2
n 1
 2.25  2.25
 4.5
1
(4) Construct the distribution of s2 by using the results of steps (2) and (3).
Solution to exercise: We can use the same probabilities for the likelihood of seeing a given
sample to fill out the table below.
Sample
(1,1)
(1,2)
(1,4)
(2,1)
(2,2)
(2,4)
(4,1)
(4,2)
(4,4)
Sample Variance, s2
0
0.5
4.5
0.5
0
2.0
4.5
2.0
0
Thus the distributions of s2 is
s2
0
0.5
2.0
4.5
Notice that
p(s2)
.44
.08
.24
.24
E[s2] = 0(.44) + (.5)(.08) + (2)(.24) + (4.5)(.24)
= 0 + .04 +.48 + 1.08
= 1.6 (!!)
Probability
.04
.04
.12
.04
.04
.12
.12
.12
.36
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